Wadler's Blog

18.8.05

The Gelfand Principle: use the simplest example; Buchberger's White-Box and Black-Box principle: learn how to execute a process in detail, then suppress the details. Together with a real-life example of how to apply them to simplify exposition.

The progress of mathematics abounds in tall tales and unlikely stories. And they don't come much more improbable than this. Outside, the July sun of the Aegean is hammering down on a coastal hotel in Mykonos. Inside, America's most charismatic statistician addresses a gathering that can boast several of the world's top mathematicians as well as a motley assortment of science writers, novelists, historians and theatre people. And what is he doing? He's performing a card trick.

Persi Diaconis, now of Stanford and Harvard Universities, once made his living this way. As a teenage prodigy, he toured the US as junior sidekick to one of the most famous magicians of the age. Then, via gamblers' after-hours talk of odds and probability, the sorcerer's apprentice caught the maths bug and took the first steps towards a career in another sort of spotlight. Diaconis was the expert who unmasked the delusions behind the so-called "Bible Codes" (which supposedly revealed hidden meanings within the text), but today in the Aegean, he's merely baffling his peers.

He chucks a deck of cards towards this highly qualified audience. It's caught by Timothy Gowers, a professor at Cambridge and recipient of a Fields Medal - the maths equivalent of a Nobel Prize. Gowers cuts the pack, takes the top card, then passes it to a neighbouring titan, who himself passes it on. After five cuts, Diaconis asks holders of red-suited cards to stand up. Two do. He then proceeds to tell all five punters exactly which card they hold. Cue a burst of awestruck applause.

How does he do it? Diaconis quips that "magicians aren't allowed to explain their secrets and mathematicians can't explain their secrets". But he tries. The root of card-recognition tricks lies in the De Bruijn Sequences, a branch of what's called "combinatorics" - a discipline with a long history that stretches from the counting patterns used in Indian classical music to the coded instructions for robots used today. The mathematicians grasp the theory easily enough, but the mind-boggling mental speed of the practice still confounds them, and me.

This is a taste ofthe first Mykonos conference on Mathematics and Narrative. Arranged by a group known as Thales and Friends, after the ancient Greek geometer and philosopher who reputedly measured the Pyramids, this unprecedented project to bring scientists and storytellers together was the brainchild of the polymath Apostolos Doxiadis. Worried that the maths he loves has drifted too far out of the cultural mainstream, Doxiadis has already done more than his share of bridge-building. His novel Uncle Petros and Goldbach's Conjecture (Faber) helps to convey the life-enhancing, and life-consuming, attraction of pure mathematical research.

Rebecca Goldstein, a philosopher and novelist who writes in her fiction about the "essentially tragic" lives of mathematicians, called her pet subjects "as bad as novelists in terms of ego". John Allen Paulos, who writes funny and instructive books, such as Innumeracy, about the misuse of statistics in the media, jokes: "How do you define an extravert mathematician? Someone who looks at your shoes when he's talking to you."

If you want evidence of the problem that confronts them, look no further than today's newspapers. Millions of people now enjoy Sudoku puzzles. Forget the pseudo-Japanese baloney: sudoku grids are a version of the Latin Square created by the great Swiss mathematician Leonhard Euler in the late 18th century. Yet these legions of amateur problem-solvers tackle puzzles accompanied by the absurd assertion that "no maths is involved". In parts of popular culture, mathematics has become not so much the love that dare not speak its name as the love that doesn't even know its name.

So, as the sun blazed and the sea sparkled off stage, we heard stories about the extraordinary rhythms of breakthrough and breakdown that punctuate the history of modern maths, and stories about the thinking and imagining that mathematicians do on the cutting edge of creation. John Barrow, another Cambridge professor, related the story of how his play Infinities reached the stage. Marcus du Sautoy, Oxford mathematician and Channel 4 pundit, delivered his multimedia gig about the mysteries of prime numbers and the long quest to prove Riemann's Hypothesis. The show took in David Beckham's Real Madrid shirt (a prime 23), some raucous audience participation and Professor du Sautoy himself on a surprisingly sweet trumpet.

Less noisily, Tim Gowers ended his plea for concreteness and compression in mathematical explanations with some favourite passages from Alan Hollinghurst, Don DeLillo and Jonathan Franzen - to highlight the skills that good novelists have and most mathematicians lack.

Of course, some writers and producers have turned to the lives and the works of mathematicians for inspiration. A gifted populariser such as Simon Singh can now sell in the hundreds of thousands - as he did with Fermat's Last Theorem. Sylvia Nasar's bestselling biography of the game-theory pioneer John Nash, and his decades-long mental illness, led to the big-screen adaptation of A Beautiful Mind. This familiar, Rain Man model of the pattern-seeking maths prodigy as a recluse, an idiot savant, or downright barking mad, recurs often - for instance, in fictionalised portraits (such as Enigma) of the computer prophet and Bletchley Park cryptographer Alan Turing. And it even underlies Mark Haddon's The Curious Incident of the Dog in the Night-time, with its Asperger-afflicted teenage narrator, always ready to reel off a series of prime numbers.

Not surprisingly, real mathematicians have mixed feelings about mass-market yarns that present their domain as the stamping-ground of eccentrics, or even lunatics. But, for the most part, they applaud the endeavour to dramatise the human struggle of mathematical reasoning. Only one (absent) literary figure really fell foul of the Mykonos mob: the American writer David Foster Wallace, who in Everything and More wrote not a novel but a purported history of the mathematics of infinity. The computer-science guru Martin Davis counted "86 really egregious errors" in Wallace's book. "Are we so hard up for approval from the humanities that we have to accept this?" he thundered.

And yet the history of modern maths features such a wealth of near-incredible narratives that certain kinds of faction or docu-drama will exert a huge appeal. After all, this is a field that, early in the last century, plunged into a "foundational crisis" that left its finest minds believing that they stood not on solid rock but on shifting sand. Out of that collective breakdown grew ideas about general computing machines that began as the purest theory but ended up as the intellectual inspiration of almost everything we now do with technology. If mathematics counts as the art of reality, then you might argue that its artistic crisis gave birth to the modern world.

This is the theme of the mathematical narrative that Doxiadis and some colleagues will tell next. Collaborating with the Berkeley-based computer scientist Christos Papadimitriou and the Athenian artists Alecos Papadatos and Annie di Donna, Doxiadis has been working on a ground-breaking graphic novel about the development of 20th-century maths and its makers, from Russell and Hilbert to Gödel and Turing.

Due in 2007, Logicomix will tell an epic human, and political, story. On the one hand, Papadatos, the project's chief graphic artist, depicts the social turmoil, global warfare and deadly ideologies of the last century. On the other, the core story of maths - as with every other brand of creativity - will often come down to the journey of a single mind alone with its dreams, and its demons. "Like a mathematician," Papadatos notes, "a cartoonist works with paper, pens - and a waste-paper basket."

LETTER TO THE EDITORMathematicians struggle for truthby Ronnie BrownThe Independent, 17 August 2005

Sir: Seeing Boyd Tonkin's article on 'Magic numbers' (15 August) I thought, as amathematician, I ought to step aside from my 'essentially tragic life', not'look at my shoes', stop 'struggling with my demons' awhile, and suggest thatperhaps a wrong impression is given of mathematics as a development of just afew strange and egocentric minds.

Instead it is a world-wide collaborative effort involving tens of thousands,struggling to understand, to see what is true and why it is true, and in sodoing to develop a language and notation for description, verification,deduction, and calculation. It describes structures and analogies. It makesdifficult things easy. So it is a basis for the modern technical world.

Mathematics can also take over for its study what Shakespeare claimed for therole of the poet: 'And as imagination bodies forth the forms of things unknown/The Poet's pen turns them to shapes, and gives to airy nothing/ A localhabitation and a name.'

All this explains its fascination, and the joy of communicating at all levels inthe subject.